Properties

Label 5040.2.a.f
Level 5040
Weight 2
Character orbit 5040.a
Self dual yes
Analytic conductor 40.245
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 5040.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 630)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{5} - q^{7} + O(q^{10}) \) \( q - q^{5} - q^{7} + 2q^{13} - 2q^{19} + q^{25} + 6q^{29} - 8q^{31} + q^{35} - 4q^{37} + 6q^{41} - 2q^{43} + 6q^{47} + q^{49} + 6q^{53} - 12q^{59} + 8q^{61} - 2q^{65} - 2q^{67} - 6q^{71} + 2q^{73} + 16q^{79} + 6q^{89} - 2q^{91} + 2q^{95} - 10q^{97} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 −1.00000 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5040.2.a.f 1
3.b odd 2 1 5040.2.a.z 1
4.b odd 2 1 630.2.a.c 1
12.b even 2 1 630.2.a.j yes 1
20.d odd 2 1 3150.2.a.bb 1
20.e even 4 2 3150.2.g.m 2
28.d even 2 1 4410.2.a.p 1
60.h even 2 1 3150.2.a.g 1
60.l odd 4 2 3150.2.g.l 2
84.h odd 2 1 4410.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.a.c 1 4.b odd 2 1
630.2.a.j yes 1 12.b even 2 1
3150.2.a.g 1 60.h even 2 1
3150.2.a.bb 1 20.d odd 2 1
3150.2.g.l 2 60.l odd 4 2
3150.2.g.m 2 20.e even 4 2
4410.2.a.p 1 28.d even 2 1
4410.2.a.y 1 84.h odd 2 1
5040.2.a.f 1 1.a even 1 1 trivial
5040.2.a.z 1 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5040))\):

\( T_{11} \)
\( T_{13} - 2 \)
\( T_{17} \)
\( T_{19} + 2 \)
\( T_{29} - 6 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + T \)
$7$ \( 1 + T \)
$11$ \( 1 + 11 T^{2} \)
$13$ \( 1 - 2 T + 13 T^{2} \)
$17$ \( 1 + 17 T^{2} \)
$19$ \( 1 + 2 T + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 - 6 T + 29 T^{2} \)
$31$ \( 1 + 8 T + 31 T^{2} \)
$37$ \( 1 + 4 T + 37 T^{2} \)
$41$ \( 1 - 6 T + 41 T^{2} \)
$43$ \( 1 + 2 T + 43 T^{2} \)
$47$ \( 1 - 6 T + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 + 12 T + 59 T^{2} \)
$61$ \( 1 - 8 T + 61 T^{2} \)
$67$ \( 1 + 2 T + 67 T^{2} \)
$71$ \( 1 + 6 T + 71 T^{2} \)
$73$ \( 1 - 2 T + 73 T^{2} \)
$79$ \( 1 - 16 T + 79 T^{2} \)
$83$ \( 1 + 83 T^{2} \)
$89$ \( 1 - 6 T + 89 T^{2} \)
$97$ \( 1 + 10 T + 97 T^{2} \)
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